Acoustic phonons and strain in core/shell nanowires

Acoustic phonons and strain in core/shell nanowires

Christoph Kloeffel Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland    Mircea Trif Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA Laboratoire de Physique des Solides, CNRS, Université Paris-Sud, 91405 Orsay, France    Daniel Loss Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
July 15, 2019
Abstract

We study theoretically the low-energy phonons and the static strain in cylindrical core/shell nanowires (NWs). Assuming pseudomorphic growth, isotropic media, and a force-free wire surface, we derive algebraic expressions for the dispersion relations, the displacement fields, and the stress and strain components from linear elasticity theory. Our results apply to NWs with arbitrary radii and arbitrary elastic constants for both core and shell. The expressions for the static strain are consistent with experiments, simulations, and previous analytical investigations; those for phonons are consistent with known results for homogeneous NWs. Among other things, we show that the dispersion relations of the torsional, longitudinal, and flexural modes change differently with the relative shell thickness, and we identify new terms in the corresponding strain tensors that are absent for uncapped NWs. We illustrate our results via the example of Ge/Si core/shell NWs and demonstrate that shell-induced strain has large effects on the hole spectrum of these systems.

pacs:
63.22.Gh, 63.22.-m, 62.20.-x, 71.70.Fk

I Introduction

In the past years, it has been demonstrated that the performance of nanowires (NWs) can greatly benefit from the presence of a shell. For instance, surface passivation is an option to reduce scattering, and measurements on InAs/InP core/shell NWs have revealed significantly higher mobilities than in uncapped InAs NWs vantilburg:sst10 (). Furthermore, experiments on various core/shell NWs treu:nlett13 (); mayer:nco13 (); hocevar:apl13 (); skoeld:nlett05 (); rigutti:prb11 () have demonstrated that adding a shell can be very useful for optical applications, a feature that is well-known, e.g., from colloidal quantum dots (QDs) mahler:nma08 (); chen:nma13 (). In Ge/Si core/shell NWs, which recently attracted attention, the shell is beneficial for several reasons. In particular, it provides a large valence band offset at the interface, leading to a strongly confined hole gas inside the Ge core without the need for dopants lu:pnas05 (); park:nlt10 ().

Due to lattice mismatch, core/shell heterostructures are usually strained, which can have important consequences on their electrical and optical properties. For instance, strain may affect the lifetimes of spin qubits maier:prb12 (); maier:prb13 () and has already been used to tune photons from separate QDs into resonance flagg:prl10 (); kuklewicz:nlett12 (). The reason for such effects lies in the strain dependence of the Hamiltonian of the electronic states birpikus:book (). For the core/shell NWs of Refs. treu:nlett13, ; skoeld:nlett05, ; hocevar:apl13, ; rigutti:prb11, , a strong and strain-based correlation between the shell thickness (or composition) and the wavelength of the emitted photons has already been measured. In addition, the shell-induced strain may lift quasidegeneracies in the spectrum of NWs and NW QDs kloeffel:prb11 (). Considering these and other possibly relevant consequences, knowledge of the strain distribution in core/shell NWs is crucial. Exact calculations of the lattice displacement, however, typically require numerics groenqvist:jap09 (); hestroffer:nanotech10 (); hocevar:apl13 (). Analytical results are rare and require simplifying assumptions that may or may not be justified, depending on the choice of materials and on the effects that one is interested in. The model of Ref. hestroffer:nanotech10, , for instance, assumes purely uniaxial strain along the NW axis. The results of Refs. liang:jap05, ; schmidt:prb08, apply when the core and shell materials are isotropic and have the same elastic properties. To our knowledge, the most general formulas provided so far are those of Ref. menendez:anphbe11, , assuming isotropic media and requiring only Poisson’s ratio to be the same in core and shell.

A particularly attractive feature of NWs is their potential to host electrically controllable spin qubits loss:pra98 (). While bare InAs and InSb NWs have been the workhorse systems fasth:prl07 (); nadjperge:nat10 (); schroer:prl11 (); vandenberg:prl13 (); kloeffel:annurev13 (), spin qubits may also be implemented in core/shell NWs such as Ge/Si hu:nna07 (); roddaro:prl08 (); hu:nna12 (); higginbotham:arX14 (), for which a large degree of external control has been predicted kloeffel:prb13 (). As electrons and holes interact with lattice vibrations, understanding of the quantum mechanical behavior of the system requires knowledge of the phonon bath. For instance, it is well-known that phonons can be dominant decay channels for spin qubits khaetskii:prb01 (); golovach:prl04 (); kroutvar:nat04 (); hanson:rmp07 (); hachiya:arX (); kornich:prb14 (). The shell of core/shell NWs not only induces static strain, it also affects the phonon modes. While phonons in homogeneous NWs have been discussed in detail in the literature cleland:book (); landau:elasticity (), we are not aware of analytical results for NWs with a finite shell.

In this paper, we derive algebraic expressions for the static strain and the low-energetic phonon modes in core/shell NWs. Assuming isotropic materials and a force-free wire surface, we allow for arbitrary core and shell radii, independent elastic properties in core and shell, and take all components of the stress and strain tensors into account. Our results for the phonons illustrate that the shell notably affects the phonon-based displacement fields and, among other things, that the dispersion relations of the longitudinal, torsional, and flexural modes change differently with the shell thickness. In particular, new terms arise in the corresponding strain and stress tensors that are absent in homogeneous NWs. We illustrate our results via the example of Ge/Si NWs, given the fact that the coherence of their interfaces has already been demonstrated experimentally dillen:prb12 (); dillen:nna14 (). The derived formulas for the static strain can be considered a further extension of those listed in Ref. menendez:anphbe11, and are consistent with experiments dillen:prb12 (); dillen:nna14 (); rigutti:prb11 (); hestroffer:nanotech10 (); treu:nlett13 (); skoeld:nlett05 (); hocevar:apl13 () and numerical simulations groenqvist:jap09 (); hestroffer:nanotech10 (); hocevar:apl13 (); rigutti:prb11 (). We calculate the effects of the static strain on the low-energy hole spectrum of Ge/Si NWs, complementing the analysis of Ref. kloeffel:prb11, .

The paper is organized as follows. In Sec. II we introduce the notation and recall relevant relations from linear elasticity theory. The results for the static strain are derived in Sec. III, where we also investigate the effects of strain on the spectrum of Ge/Si NWs. Having summarized the low-energetic phonon modes in homogeneous NWs in Sec. IV, we extend these results to the case of core/shell NWs in Sec. V, followed by concluding remarks in Sec. VI. The appendix contains useful relations and further details of the calculations. In particular, providing also a comparison to the case of bulk material, we discuss the displacement operator and the normalization condition for phonons in core/shell and core/multishell NWs, as quantization is mandatory for quantum mechanical analyses.

Ii Linear elasticity theory

In this section we recall relevant relations from linear elasticity theory and introduce the notation used throughout this paper. The information summarized here is carefully explained in Refs. cleland:book, ; landau:elasticity, , and we refer to these for further details.

In a bulk semiconductor without additional forces, the atoms form a periodic and very well structured lattice, characterized by the lattice constant . The displacement of an atom from its original position is described by the displacement vector . It may be caused by externally applied forces, or, as in the case of core/shell NWs, by an interface between materials with different lattice constants. In the continuum model, the displacement field is directly related to the strain tensor elements via

(1)

leading to strain-induced effects on the conduction and valence band states birpikus:book (). The position is denoted here by , where the three orthonormal basis vectors are the unit vectors along the orthogonal axes . Important quantities besides the strain are the stress tensor elements . These are of relevance as corresponds to the force along experienced by an area normal to . We note that and , which implies that the strain and stress tensors are fully described by three diagonal and three off-diagonal elements each.

For semiconductors with diamond (Ge, Si, …) or zinc blende (GaAs, InAs, …) structure, the relation between stress and strain is given by

(2)

where the are the elastic stiffness coefficients and are the main crystallographic axes. Calculations with the exact stiffness matrix often require elaborate numerical analyses, and it is therefore common to replace the stress-strain relations by those of an isotropic material. This simplification usually results in good approximations when compared with the precise simulations cleland:book (); landau:elasticity (). The elastic properties of such a material are fully described by the two Lamé parameters and , which are found from Young’s modulus (often denoted by ) and Poisson’s ratio through

(3)
(4)

Considering the limit of isotropic media, we thus replace the stiffness coefficients of Eq. (2) by , , and . The relations between stress and strain are now the same for arbitrarily rotated coordinate systems. Hence, referring to NWs, we obtain

(5)

where

(6)
(7)

and are the orthonormal basis vectors for cylindrical coordinates , , and . The vector is oriented along the symmetry axis of the NW, while and point in the radial and azimuthal direction, respectively. From

(8)

it is evident that the cartesian coordinates and are related to and through and (the axis is the same in both coordinate systems). We wish to emphasize that we use

(9)

throughout this work in order to avoid confusion with the density , and so . Detailed information about the stress and strain tensor elements in cartesian and cylindrical coordinates is provided in Appendixes B.2 and B.3. Finally, we note that Eq. (5) is independent of the growth direction of the NW because of the isotropic approximation.

Iii Static strain in core/shell nanowires

An interface between two materials of mismatched lattice constants induces a static strain field. In core/shell NWs, such an interface is present at the core radius . When the lattice constants in core () and shell () are different, the system will tend to match these for reasons of energy minimization. For instance, and for Ge/Si core/shell NWs winkler:book (); adachi:properties (), and so the shell tends to compress the core lattice, strongly affecting the properties of the confined hole gas kloeffel:prb11 (); birpikus:book (). Below, we analyze the strain in core/shell NWs and derive algebraic expressions for both the inner and outer part of the heterostructure. The resulting static strain is found by assuming a coherent interface between the two materials, i.e., pseudomorphic growth. We consider the limit of an infinite wire, which applies well away from the ends of the NW groenqvist:jap09 () when the length is much larger than the shell radius (). Our approach is similar to those used previously liang:jap05 (); schmidt:prb08 (); groenqvist:jap09 (); hestroffer:nanotech10 (); menendez:anphbe11 ().

iii.1 Boundary conditions

When the strain changes slowly on the scale of the lattice spacing, the displacement field leads to the distorted lattice vectors groenqvist:jap09 ()

(10)

when viewed from an atom at position in the core () or shell (), respectively, where are the orthonormal basis vectors of the lattice, is a vector with integer components , and is the Kronecker delta. Pseudomorphic strain requires the components of the distorted lattice vectors in core and shell that are parallel to the interface to match at the core-shell transition. Thus, , where stands for an arbitrary tangent to the core-shell interface. Using cylindrical coordinates, the two orthogonal directions and are the basis vectors for any , which results in the boundary conditions and at radius . Furthermore, the core-shell transition needs to be spatially matched in the radial direction, i.e., there should be no unrealistic gaps or overlaps between the two materials at the interface.

In order to ensure pseudomorphic strain, we start from an initial configuration where the shell is unstrained and the core is highly strained, such that the lattice constant of the core matches the one in the shell povolotskyi:jap06 (); groenqvist:jap09 (). Of course, this initial arrangement is unstable, and so the system will relax into a stable and energetically favored configuration. Considering the continuum limit, the requirements for a coherent interface at can now be summarized in a simple form nishiguchi:prb94 (),

(11)

where the , in contrast to the of Eq. (10), denote the displacement from the initially matched configuration. That is,

(12)

and . As illustrated in Eq. (12), the displacement field in the core can be described by a sum of two parts. When the lattice constant changes from to , the term shifts an atom that is originally located at to its new position . Additional displacement from this new position is then accounted for by . Consequently, the strain tensor elements in the core and shell read

(13)

where . The resulting strain tensor is linearly related to the stress tensor via the Lamé constants [Eq. (5)]. As additional boundary conditions, the stress must be continuous at the interface nishiguchi:prb94 () and we assume that the shell surface is free of forces,

(14)
(15)

Next, using the above-mentioned boundary conditions, we derive algebraic expressions for the static strain in core/shell NWs.

iii.2 Analytical results

From symmetry considerations, the displacement in both core and shell must be of the form

(16)

where we introduced for convenience. For the displacement field to be static, the differential equations

(17)

need to be solved in the absence of body forces landau:elasticity (), and in doing so we find

(18)
(19)

where and are the Lamé parameters in the core and shell, respectively. The coefficients to are to be determined from the boundary conditions. Since the displacement must be finite in the center, we first conclude that . Second, also because must vanish at the surface [Eq. (15)], and so the are constants. Consequently, one obtains

(20)

with as the resulting (equilibrium) lattice constant in the direction. From the boundary conditions listed in Eqs. (11), (14), and (15), we can express all nonzero coefficients

(21)
(22)

and in terms of only. The latter can finally be found by minimizing the elastic energy of the system.

Using Eq. (13), the above-mentioned results for , and the equations of Appendix B.3 for the strain tensor in cylindrical coordinates, one finds

(23)
(24)

in the core, and

(25)
(26)
(27)

in the shell, with . The parameter

(28)

introduced in Eqs. (23) and (24) is the relative mismatch of the lattice constants. Remarkably, Eqs. (23) and (24) imply that the stress and strain are constant within the core, which is consistent with simulations groenqvist:jap09 (); hocevar:apl13 (); hestroffer:nanotech10 (). Furthermore, we note that . Below, we outline the calculation of and provide our final results.

The elastic energy density in an isotropic solid is landau:elasticity ()

(29)

where is the trace of the strain tensor, and

(30)

is the elastic energy of an object with volume . For the static strain discussed in this section, the elastic energy density in the core is constant and in the shell depends solely on the coordinate . The elastic energy of the NW can therefore be calculated via

(31)

By imposing the condition in order to find the energetically favored configuration, we obtain algebraic expressions for and, thus, for all previously discussed quantities. As expected, these are not affected by the length of the wire, since for the regime considered here. Moreover, the coefficients and do not depend on the absolute values of and . Instead, they depend on the relative shell thickness

(32)

More precisely, it is possible to write the dependence of and on the radii in terms of

(33)

only, which is the ratio between the shell and core area in the cross section. Similarly, the coefficients , , and do not depend on the absolute values of the lattice constants and , but on the relative lattice mismatch [Eq. (28)]. We note that , which is important for the linear elasticity theory of this work to hold.

The full results are rather lengthy. Nevertheless, they can be very well approximated through an expansion in the small parameter . Neglecting corrections of order and rewriting the results in a convenient form, we obtain

(34)
(35)
(36)

where we defined

(37)

The denominator is

(38)

Inserting , , and the above-listed expressions into Eqs. (23) and (24), and neglecting again terms of order , one finds

(39)
(40)

for the strain in the core.

The expressions derived in this work are more general than those provided previously hestroffer:nanotech10 (); liang:jap05 (); schmidt:prb08 (); menendez:anphbe11 () and may be interpreted as a further extension of those in Ref. menendez:anphbe11, . Indeed, by writing the Lamé parameters and in terms of Young’s modulus and Poisson’s ratio for core and shell [Eqs. (3) and (4)], we find that our results are exactly identical to those of Ref. menendez:anphbe11, for the case assumed therein. From the ratio of Eqs. (39) and (40),

(41)

we find that typically , in agreement with numerical results hocevar:apl13 (); hestroffer:nanotech10 (); rigutti:prb11 (); groenqvist:jap09 (). Our formulas also feature the correct limits. For instance, we obtain for as expected, and so the strain in the core vanishes for a negligibly thin shell. In the limit of an infinite shell (), our formulas yield , , and

(42)

which corresponds exactly to the previously studied case of a wire embedded in an infinite matrix yang:prb97 (). When switching to cartesian coordinates, we obtain , which seems consistent with numerics groenqvist:jap09 (). As pointed out in Ref. groenqvist:jap09, , where core/shell NWs with anisotropic materials have been investigated, also other off-diagonal strain tensor components may be nonzero in reality. However, these were found to be very small, particularly in the core.

iii.3 Results for Ge/Si core/shell nanowires

We conclude our discussion of the static stress and strain fields by applying our results to the example of Ge/Si core/shell NWs, demonstrating that the strain can have major effects on the electronic properties of a system. As illustrated in this example, the strain is usually not negligible, and so our formulas derived for both core and shell may prove very useful for a wide range of material combinations.

Ge/Si core/shell NWs have attracted attention because they host strongly confined hole states inside their cores lu:pnas05 (). Thus, we focus here on the static strain in the core and discuss its effects on the holes in more detail. The lattice mismatch for Ge/Si core/shell NWs is winkler:book (); adachi:properties (), and the Lamé constants, listed in units of , are , , , and (see also Appendix A) cleland:book (); adachi:properties (). In Fig. 1 (top), the strain tensor elements and of Eqs. (39) and (40) are plotted as a function of . These are negative over the entire range of shell thicknesses, and so the core material is compressed, as expected from . The dependence of the strain on is consistent with simulations and experiments hestroffer:nanotech10 (), and we note that for any .

Figure 1: Static core strain and its effect on the hole spectrum of Ge/Si core/shell NWs as a function of the relative shell thickness . In the top figure, the nonzero components and of Eqs. (39) and (40) are plotted for the parameters in the text. The resulting splitting [Eq. (47), bottom figure] can be as large as 30 meV and strongly affects the low-energetic hole states, as explained in detail in Ref. kloeffel:prb11, [wherein ].

The effects of strain on hole states in the topmost valence band of Ge are described by the Bir-Pikus Hamiltonian birpikus:book (). Using the spherical approximation , which applies well to Ge (, birpikus:book ()), and neglecting global shifts in energy, the Bir-Pikus Hamiltonian for holes reads

(43)

Here and are the deformation potentials, are the components of the effective spin 3/2 along the axes , “c.p.” stands for cyclic permutations, and . We note that the axes , , and need not coincide with the main crystallographic axes due to the spherical approximation. Our results for the static strain in core/shell NWs reveal that the relations and are fulfilled in the core. Exploiting these properties and the equality , the Bir-Pikus Hamiltonian for the core is of the simple form

(44)

where global shifts in energy have again been omitted. As discussed in Ref. kloeffel:prb11, , this Hamiltonian has important effects on the hole spectrum in Ge/Si core/shell NWs, because it determines the splitting between the ground states and the first excited states at wave number along the NW. The subscripts “” and “” refer to the spin states, and we note that the total angular momentum along the wire is for , whereas for . The splitting comprises a strain-independent term , which arises from the radial confinement and the kinetic energy (Luttinger-Kohn Hamiltonian), and the strain-induced term

(45)

Defining

(46)

one finally obtains

(47)

The parameter turns out to be independent of , and using and of Ref. kloeffel:prb11, we find for Ge. That is, is determined by and depends only on the relative shell thickness.

Figure 1 (bottom) shows the dependence of on for Ge/Si core/shell NWs. Remarkably, can be as large as 30 meV and exceeds at relatively thin shells () already. For comparison, one finds for typical core radii . Therefore, the splitting is mostly determined by , i.e., by the relative shell thickness. The combination of a small and is of great importance not only for the spectrum in the wire, but also for, e.g., the properties of hole-spin qubits in NW QDs kloeffel:prb11 (); maier:prb13 (); kloeffel:annurev13 (); kloeffel:prb13 ().

Iv Phonons in homogeneous nanowires

In this section, we recall the calculation of lattice vibrations in homogeneous NWs landau:elasticity (); cleland:book (); stroscio:book (); trif:prb08 () and provide the displacement vectors and the core strain for the phonon modes of lowest energy. The information forms a basis for Sec. V, where the analysis is extended to the case of core/shell NWs.

iv.1 Equation of motion, ansatz, and boundary conditions

For an isotropic material with density and Lamé parameters and , the equations of motion

(48)

can be summarized in the form

(49)

where

(50)

is the Nabla operator and

(51)

is the Laplacian. In order to find the eigenmodes for the cylindrical NW, the displacement vector may be written in terms of three scalar functions , , via stroscio:book (); nishiguchi:prb94 (); sirenko:pre96 ()

(52)

Inserting Eq. (52) into Eq. (49) and exploiting identities such as , the resulting equation of motion reads

(53)

This equation is therefore satisfied when the scalar functions obey the wave equations

(54)

where is a Kronecker delta. While these wave equations are sufficient criteria for the equation of motion to be satisfied, we note that some special solutions of Eq. (49) can be found that do not obey Eq. (54). An example is provided below for the torsional mode. However, we also illustrate that this solution can be interpreted as the limit of a more general solution obtained with an ansatz that relies on the above-mentioned wave equations.

Due to the cylindrical symmetry and the translational invariance along the NW axis ( radius), the can be written in the form

(55)

where is the wave number along the wire, is an integer, is the angular frequency, and is the time. Insertion of Eq. (55) into Eq. (54) results in the differential equation

(56)

for the function . With and as Bessel functions of the first and second kind, respectively, and with and as (dimensionful) complex coefficients, the general solution of this differential equation is

(57)

where

(58)

We mention that and need not be identical and may be chosen arbitrarily, provided that Eq. (58) is satisfied. For homogeneous NWs considered in this section, because diverges in the limit , and so

(59)

For given and , the corresponding eigenfrequencies and coefficients can be determined from the boundary conditions that we discuss below.

In vector notation, with the three components referring to , , and , respectively, Eq. (52) reads as

(60)

From Eq. (5) and the equations in Appendix B.3, one finds

(61)

for the stress related to the radial direction. For a homogeneous NW of radius , the boundary conditions are

(62)

i.e., the stress tensor elements , , and must vanish at due to the assumption of a force-free wire surface. Using the ansatz introduced above, these boundary conditions can be written in the form

(63)

where are vectors with components , , and . The boundary conditions can only be met in a nontrivial fashion (i.e., not all are zero) when the corresponding determinant vanishes,

(64)

For given and , the allowed angular frequencies can be found from this determinantal equation. We note, however, that a root of Eq. (64) does not necessarily correspond to a physical solution that describes a phonon mode. The latter can be found for given , , and by calculating the coefficients from the set of boundary conditions. One of these coefficients can be chosen arbitrarily and determines the phase and amplitude of the lattice vibration. In a quantum mechanical description, this coefficient is finally obtained from the normalization condition.

There are four types of low-energetic phonon modes in a NW: one torsional (; ), one longitudinal (; ), and two flexural modes (; ). These modes are referred to as gapless, as their angular frequencies and, thus, the phonon energies converge to zero when . In the following, we summarize the dispersion relation, the displacement field, and the strain tensor elements for each of these modes. We consider the regime of lowest energy, i.e., the regime of small for which an expansion in applies. We note that the investigated phonon modes are acoustic modes, as the atoms of a unit cell move in phase, i.e., in the same direction, in contrast to the out-of-phase movement of optical phonons where the atoms of a unit cell move in opposite directions.

iv.2 Torsional mode

We start our summary with a special solution. It can easily be verified that the displacement cleland:book ()

(65)

meets the boundary conditions, as . The prefactor is a dimensionless complex number and may be chosen arbitrarily. Furthermore, defining angular frequencies as positive, the equation of motion [Eq. (49)] is satisfied for

(66)

Due to the displacement along , this mode is referred to as torsional (), and is the speed of the corresponding sound wave. From Eq. (60), it can be seen that of Eq. (65) is generated via and

(67)

A special feature of this result compared to the others summarized in this work is that does not obey the wave equation, Eq. (54). Moreover, the presented solution for the torsional mode in homogeneous NWs is exact and does not require an expansion in . The only nonzero strain tensor element in cylindrical coordinates is

(68)

and we mention that and in cartesian coordinates.

In Sec. V.2, the torsional mode in core/shell NWs is investigated with an ansatz based on Bessel functions, for which Eq. (54) is satisfied. It is therefore worth mentioning that the special solution for homogeneous NWs is obtained in the limit of a vanishing shell. We consider and

(69)

as an ansatz, assuming , i.e., , which may be due to the presence of a shell. The resulting displacement function is

(70)

and we note that . The arbitrary coefficient may be written as . Considering as a small parameter, expansion yields

(71)

and

(72)

For , i.e., , which corresponds to the limit of a vanishing shell, one finds that the boundary condition of a force-free wire surface is fulfilled due to . As anticipated, the displacement converges to the solution for homogeneous NWs, Eq. (65).

iv.3 Longitudinal mode

The longitudinal and torsional modes in the NW have no angular dependence, . In stark contrast to the torsional mode, however, the longitudinal mode () does not lead to displacement along , and so . The boundary condition is therefore fulfilled and one may set in the ansatz discussed in Sec. IV.1. Analogously to Eq. (64), the eigenfrequencies can be calculated via the determinant of a 22 matrix that summarizes the remaining boundary conditions. Considering angular frequencies as positive, one finds that the dominant terms of this determinant vanish for landau:elasticity (); cleland:book (); sirenko:pre96 ()

(73)
(74)

where is the corresponding speed of sound. The properties of the longitudinal and flexural modes can conveniently be written in terms of Young’s modulus and Poisson’s ratio [see also Eqs. (3) and (4)]

(75)
(76)

Introducing the dimensionless as an arbitrary complex prefactor, the resulting displacement vector is of the form

(77)

with , , and as the basis vectors. Here and in the remainder of the section, refers to higher-order terms of type , where and are integers. For the strain tensor elements in cylindrical coordinates, one obtains

(78)
(79)